For requesting, clarifying, and comparing definitions of mathematical terms.

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100 views

Path-independent contour integrals and how to define them

If a contour $C$ is parameterized by $z(t): [a, b] \to \mathbb{C}$, then we define $$ \int_C f(z) \, dz= \int_a^b f(z(t)) \, z'(t) \, dt.$$ If the contour integral on the left side is equal to some ...
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75 views

Types of definitions

My question is about terminology. Consider the following two types of definitions of a circle: A circle is the collection of all points at the same distance from a given point. A circle is the ...
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35 views

Meaning of proper antichain

I saw this sequence on the Online Encyclopedia of Integer Sequences, which describe the number of 3-element proper antichains of an n-element set. What does it mean to be a 3-element proper antichain ...
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65 views

Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
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126 views

Confused about the definitions of atom

I went though half a dozen books on measure theory, and it occurs to me that the definition of atom is not particularly unified. Version $1$: A set $E$ in a $\sigma$-algebra $\Sigma$ is called ...
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128 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
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2k views

Proper functions

I'm a bit confused with the definition of proper function, i.e.: "$f: X \to Y$ is proper if the inverse image of every compact set in $Y$ is compact in $X$." Can anyone provide a more intuitive ...
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59 views

Is the empty matrix $M$ ($m \times n$) with $m$ or $n$ equal to zero in RREF ? Does it posses every matrix property one would wish?

Is the empty matrix $M$ ($m \times n$) with $m$ or $n$ equal to zero in RREF ? Reading a proof in Linear Algebra induction starts off by proving $P(n)$ holds for $n=0$, where $P(n)$ is the statement:...
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117 views

Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, \...
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107 views

What is a Line Integral?

My question is very simple yet crucial to the understanding of many fields of mathematics. What is a line integral? If I choose some arbitrary line segment $\mathbb{A}$ to integrate a function $\Gamma$...
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122 views

Meaning of quasiperiodicity in classical KAM

I'm learning about the classical KAM theorem, and I can't quite infer precisely what the term "quasi-periodic solution" means in the theorem's statement. I'm reading the following introductory note: ...
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212 views

Are edge cuts, vertex cuts, and cut sets all variously called “cuts”?

I've seen "cut" being used to refer to all three, in different places, and sometimes in the same book. Which does "cut" most commonly refer to? p.s. I am aware that "cut" itself can be defined to ...
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64 views

Can a sample space depend on the parameter to be estimated (example: # of cabs in a city)

In our intro statistic lecture the following we said that the following components made up an estimation problem an at most countable space $\mathcal{X}$ of all possible samples we can observe a ...
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94 views

Graph Theory: Help with a definition

I need some help to see if the definition I found of Cycles and Cycles Decomposition is right, here it is: A graph is a Cycle if it is isomorph to another graph $G=(V,E)$ with the following ...
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434 views

Explanation of Mixed Strategy Definition in Game Theory

Definition: Let $(N, A, u)$ be normal-form game, and for any set $X$ let $\Pi(X)$ be the set of all probability distributions over $X$. Then the set of fixed strategies for player $S_i=\Pi(A_i)$. ...
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84 views

Name for the initial object of a connected component of a category

I was just wondering whether there is already a name for an object of a category as described below. Recall that an initial object in a category $\mathcal C$ is an object $I\in\mathcal C$ such that ...
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94 views

Where can I find a description of math language symbols?

I am reading math articles. I meet math symbols. For example $\exists$ or $\forall$. For example for "For any a exist e that" can be rewriten as: $\forall a \exists e$ Where can I find full ...
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224 views

How to prove that $e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt[n]{n\#} $?

While reading this post, I stumbled across these definitions (Wiki_german) $$e = \lim_{n \to \infty} \sqrt[n]{n\#}$$ and $$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$$ The last one seems ...
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57 views

terminology clarification needed: projection of a measure to an open set

I'm reading in Doobs "Classical Potential Theory and its Probabilistic Counterpart" and I'm having trouble with terminology. Specifically, in Part I chapter 6, he talks about the projection of a ...
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105 views

question about definition Silverman's AEC

I am sure many of you have Silverman's book the arithmetic of elliptic curves (2nd edition). On page 417, proposition 1.2, he defines $\delta$. He also defines $\xi: G \rightarrow M:\ \xi_\sigma=m^{\...
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284 views

why are curl and divergence defined the way they are?

Curl of a given vector field, $F$, at a given point is a vector, $C$, that gives the measure of amount of rotation the vector undergoes at that point. It has been defined using cross products and ...
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16 views

Arithmetic-definability of geometrically-defined arithmetic concepts

Arithmetic-definability of geometrically-defined arithmetic concepts For purposes of discussion take arithmetic to be the study of the [ positive real] numbers, sequences of numbers, etc. and take ...
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15 views

Finitely generated module: terminology.

What's the meaning of the expression: $S$ is a subring of $\mathbb{C}$ finitely generated as $\mathbb{Z}$-module? Maybe that the additive group of the ring $S$ is a finitely generated abelian group? ...
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49 views

What is the formal definition of repeated limit?

The basic question is what has been asked in the title. I looked for the definition here, here and here but no definition uses quantifiers. I tried to formulate the definition but succeeded only ...
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20 views

Definition of the left and right derivative.

The definition of the derivative is $$g'(a)=\lim \limits_{\delta \rightarrow 0} \frac{g(a+\delta) - g(a)}{\delta}$$ also the left derivative is $$ \lim \limits_{\delta \rightarrow 0^-} \frac{g(a+\...
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24 views

Arc-bases and Point-bases: when are they different for finite digraphs?

Definitions Given a digraph $D=(X,U)$, a point-basis of a digraph $D$ is a minimal vertex set from which a dipath exist to every vertex in $D$. An arc-basis is a minimal arc-reaching set. A subset ...
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24 views

definition of a traveling wave (solution)

today I've got a problem with understanding what is meant by the term "traveling wave solution" of a PDE in a more general sense. To be more precise, I've got a following Poisson equation \begin{...
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37 views

Relevancy of standard deviation

I'm a maths student but I'm a bit of a retard when it comes to probability. I have this very basic question in mind : Why is standard deviation defined as the square root of the variance, when surely ...
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28 views

What are the components and quasicomponents?

I'm trying to understand the difference between components and quasicomponents. I'm using the following definitions: $x\sim y$ iff $x$ and $y$ lie together in some connected set. The component of ...
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22 views

Are these correct extension of injective and surjective functions?

Recall the definition of injective and surjective functions: 1.a A function $f: X \to Y$, is injective if for all $a,b \in X$, if $a \neq b$, then $f(a) \neq f(b)$ 1.b A function $f: X \to Y$,...
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50 views

Locality and base change along effective descent morphisms

Definition. Let $\mathcal M$ be a class of arrows in a site $\mathsf{C}$. An arrow $\pi:E\rightarrow B$ is said to be locally in $\mathcal M$ if there's a covering ${u_i:U_i\rightarrow B}$ such that $...
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59 views

Definition of regular map from aquasi projective variety

These are from the book Basic Algebraic geometry by Shafarevich Definition of regular function on a quasi projective variety is as follows : Let $X\subset \mathbb{P}^n $ be a quasi projective ...
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29 views

Clarification of Sperner's Lemma

From Graph Theory by Bondy, Murty Image from wikipedia I don't see how the picture holds according to the definition from the Graph Theory book. Specifically, the definition says to assign ...
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45 views

Projective and affine varieties: differences, advantages and why two definitions

I have recently started to learn algebraic geometry and this question has been bugging me. An affine variety is a zero set of a collection of polynomials in affine space and a projective variety is ...
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49 views

Meaning of alternation in definition of wedge product

Spivak defines the wedge product as $\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$ and I have been running into some conceptual issues here. The alternation is defined as $...
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27 views

w.r.t. which chain complex is $H^k_{sign}(M;R)$ computed?

This question is inspired by my previous question. People often write $H^k_{sign}(M;R)$ for the $k$th singular cohomology group with coefficients in $R$. However, I don't understand what this means. ...
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43 views

How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. ...
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20 views

Is a finite dimensional vector also a covector?

I am reading something where it seems they have defined a covector/linear functional as a map $F: \mathbb{R}^n \to \mathbb{R}$ where $F$ is a vector For example, $F(x) = \begin{bmatrix} 1 & 2 \...
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33 views

$\mathsf{Top}^2$ is a stack over $\mathsf{Top}$?

In section 4.1 of Vistoli's notes he starts by showing we may locally construct arrows in $\mathsf{Top}^2$, i.e arrows over $U$. Then, the author says that we can moreover do the same for spaces, ...
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43 views

different definitions of a group

A group is classically defined as a set with a binary operation (the group product) which is associative, such that there is a unit and for every element there is an inverse. I know we can define a ...
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68 views

Is there an accepted definition of coprimality in commutative ring theory?

I can think of at least three possible definitions of coprimality in commutative ring theory: call $a,b \in R$ are coprime iff if $c \mid a$ and $c \mid b$, then $c \mid 1$. if $a \mid c$ and $b \...
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30 views

Étale morphisms definition?

Working over a commutative ring $R$, let $D= \left\{ d\in R : d^2 =0 \right\}$ A formally étale morphism $f:M\rightarrow N$ is one for which the square below is a pullback for every point $d:\mathbf{1}...
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45 views

Is this how we define “limit in the distributional sense”?

Consider $\mathcal{D}(\mathbb{R})$ the space of test functions and $\mathcal{D}'(\mathbb{R})$ the space of distributions, in $\mathbb{R}$, i.e., continuous linear functionals over $\mathcal{D}(\mathbb{...
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70 views

What is the name of the property $x^m=x$ when $x$ is in a ring?

I have been doing problems in Atiyah & MacDonald's Introduction to Commutative Algebra, and in problem 1.6 it asks to assume the existence of an idempotent element in an ideal whenever the ideal ...
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63 views

How are non-associative groups called?

Is there a name for the algebraic objects that have all the properties of groups expect associativity? For example, the unit octonions have this property. They satisfy the following definition. ...
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51 views

Does there exist a derivative of the Golden Ratio (equation)?

On my calculus exam, there was a question asking "Is the Golden Ratio differentiable? If so, use the definition of a derivative to show it." The "definition of a derivative" is $f'(x)=\displaystyle\...
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16 views

Different meanings of $L_r(P)$: norm, metric, class of functions

I have doubts on the meaning of the symbol $L_r(P)$ for $r\in \mathbb{N}$. Consider a random variable $X: \Omega \rightarrow \mathbb{R}$ with probability distribution $P$ and a random function $f:\...
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55 views

What does non-degenerate mean?

The context is that f is a real-valued function on a plane such that for every non-degenerate square ABCD in the plane, f(A)+f(B)+f(C)+f(D)=0.... What does this word mean here? I've found that "in ...
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39 views

Meaning of “Form” in mathematics

There are a number of occasions in the mathematics where the word "form" is used: Modular form Bilinear form Quadratic form ... It seems that form is a special kind of function. But I cannot ...
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22 views

Ideal Sheaf: Making sure I got it right

I want to know if I unwrapped the definition of ideal sheaves correctly. Let $\mathscr{O}$ be a sheaf of rings on topological space $X$. The definition of an ideal sheaf is as follows: The ideal ...